Constructing various simple polygonal tensegrities by directly or recursively adding bars

As an important type of tensegrities, polygonal tensegrities and their assemblies/variants hold various applications in, such as, architectures, robotics, and metamaterials. We here propose two methods (direct and recursive methods) to construct the topologies of simple polygonal tensegrities by directly and recursively adding bars, respectively. In the design of a simple 2n-polygonal topology using the direct method, we arrange 2n strings and 2n nodes according to the edges and vertices of simple 2n-polygon, and then add n bars between each two of un-neighboring nodes. In the same topological design using the recursive method, we construct the topology of quadrilateral tensegrity as a start, then add one bar to generate a hexagonal topology, and repeat this operation until the total number of added bars reaches n. These two methods yield exactly the same configurations, showing that the total number of simple polygonal topologies increases exponentially with the number of bars. Surprisingly, there exist almost 100 million simple 22-polygonal topologies. Finally, we validate both numerically and experimentally that each of our designed topologies can produce a stable tensegrity configuration without external loads. This work sheds light on the topological features and mechanical characteristics of polygonal tensegrities, which help broaden their application scenarios.